Quadratically convergent — doubles correct digits each iteration. Reaches hundreds of digits in ~10 steps. The fastest converging algorithm here.
Quadratic convergenceUses arctangent identities: π/4 = 4·arctan(1/5) − arctan(1/239). Linearly convergent, but very fast per digit — used for record calculations historically.
Linear — fast seriesAn improvement on Leibniz: π = 3 + 4/(2·3·4) − 4/(4·5·6) + … Converges much faster than Leibniz but still slowly versus Machin or GL.
Cubic convergenceThe classic: π/4 = 1 − 1/3 + 1/5 − 1/7 + … Beautiful but painfully slow — needs millions of terms for just a few correct digits.
Extremely slow